Harmonic bases for generalized coinvariant algebras

Let $k \leq n$ be nonnegative integers and let $λ$ be a partition of $k$. S. Griffin recently introduced a quotient $R_{n,λ}$ of the polynomial ring $\mathbb{Q}[x_1, \dots, x_n]$ in $n$ variables which simultaneously generalizes the Delta Conjecture coinvariant rings of Haglund-Rhoades-Shimozono and the cohomology rings of Springer fibers studied by Tanisaki and Garsia-Procesi. We describe the space $V_{n,λ}$ of harmonics attached to $R_{n,λ}$ and produce a harmonic basis of $R_{n,λ}$ indexed by certain ordered set partitions $\mathcal{OP}_{n,λ}$. The combinatorics of this basis is governed by a new extension of the {\em Lehmer code} of a permutation to $\mathcal{OP}_{n, λ}$.

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